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-2t^{2}+12t=\frac{9}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
-2t^{2}+12t-\frac{9}{2}=\frac{9}{2}-\frac{9}{2}
Subtract \frac{9}{2} from both sides of the equation.
-2t^{2}+12t-\frac{9}{2}=0
Subtracting \frac{9}{2} from itself leaves 0.
t=\frac{-12±\sqrt{12^{2}-4\left(-2\right)\left(-\frac{9}{2}\right)}}{2\left(-2\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 12 for b, and -\frac{9}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
t=\frac{-12±\sqrt{144-4\left(-2\right)\left(-\frac{9}{2}\right)}}{2\left(-2\right)}
Square 12.
t=\frac{-12±\sqrt{144+8\left(-\frac{9}{2}\right)}}{2\left(-2\right)}
Multiply -4 times -2.
t=\frac{-12±\sqrt{144-36}}{2\left(-2\right)}
Multiply 8 times -\frac{9}{2}.
t=\frac{-12±\sqrt{108}}{2\left(-2\right)}
Add 144 to -36.
t=\frac{-12±6\sqrt{3}}{2\left(-2\right)}
Take the square root of 108.
t=\frac{-12±6\sqrt{3}}{-4}
Multiply 2 times -2.
t=\frac{6\sqrt{3}-12}{-4}
Now solve the equation t=\frac{-12±6\sqrt{3}}{-4} when ± is plus. Add -12 to 6\sqrt{3}.
t=-\frac{3\sqrt{3}}{2}+3
Divide -12+6\sqrt{3} by -4.
t=\frac{-6\sqrt{3}-12}{-4}
Now solve the equation t=\frac{-12±6\sqrt{3}}{-4} when ± is minus. Subtract 6\sqrt{3} from -12.
t=\frac{3\sqrt{3}}{2}+3
Divide -12-6\sqrt{3} by -4.
t=-\frac{3\sqrt{3}}{2}+3 t=\frac{3\sqrt{3}}{2}+3
The equation is now solved.
-2t^{2}+12t=\frac{9}{2}
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{-2t^{2}+12t}{-2}=\frac{\frac{9}{2}}{-2}
Divide both sides by -2.
t^{2}+\frac{12}{-2}t=\frac{\frac{9}{2}}{-2}
Dividing by -2 undoes the multiplication by -2.
t^{2}-6t=\frac{\frac{9}{2}}{-2}
Divide 12 by -2.
t^{2}-6t=-\frac{9}{4}
Divide \frac{9}{2} by -2.
t^{2}-6t+\left(-3\right)^{2}=-\frac{9}{4}+\left(-3\right)^{2}
Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
t^{2}-6t+9=-\frac{9}{4}+9
Square -3.
t^{2}-6t+9=\frac{27}{4}
Add -\frac{9}{4} to 9.
\left(t-3\right)^{2}=\frac{27}{4}
Factor t^{2}-6t+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(t-3\right)^{2}}=\sqrt{\frac{27}{4}}
Take the square root of both sides of the equation.
t-3=\frac{3\sqrt{3}}{2} t-3=-\frac{3\sqrt{3}}{2}
Simplify.
t=\frac{3\sqrt{3}}{2}+3 t=-\frac{3\sqrt{3}}{2}+3
Add 3 to both sides of the equation.